27-Feb-2017: Verifying a power law for an inertial pendulum

When we want to detect the weight of mass, we can use a spring or a machine to measure it. However, is there any other way to find out the weight without using those "old" items? The following gives us a new way to measure the mass.

Purpose: Finding a relationship between mass and period.

In the beginning, Professor Wolf gave us a function T = A(m + Mtray)^n  ( m is the mass we added, Mtray is the weight is the tray, T is the period, A and n are unknown constant. It turned our purpose into verifying the power law.)

Set up:

1. Use a C-clamp to secure the inertial balance to the tabletop. Put a thin piece of masking tape on the end of the inertial balance.
2. Set up a photogate so the when the balance is oscillating the tape completely passes through the beam of the photogate. 
   
   
   
3. Set up the LabPro with a power adapter, USB cable, and plug adapter plugged into the DIG/SONIC1 input. 
4. Open the Logger Pro application. 
   
5. Then, start to collect the data.
   

Analyze:

1. We can easily simplify the function T = A(m + Mtray)^n by using nature log. Then, we get lnT = n•ln(m + Mtray) + lnA. 
2. So the function turns into a linear function, which is y = kx + b. 
3. Because we do not know the mass of tray, we can suppose it into different value to make the function seem to be a straight line. 

First, we tried Mtray = 220g

We get correlation = 0.9986

Second we tried Mtray = 260g

We get correlation = 0.9994

Last we tried Mtray = 320g

We get correlation = 0.9994

However, this result is a little bit different from what we get in Logger Pro.
In Logger Pro, when we use Mtray = 260g, we get correlation = 0.9999

At that time, we assumed Mtray is around 260g.
In that situation, we know lnT = 0.6246•ln(m + 260) - 4.726

Extension:

• After that, we found the smallest value and the largest value for Mtray when correlation still equal to 0.9999 by changing its value. We get 253g and 280g.
• Before the end, we used the function lnT = n•ln(m + Mtray) + lnA to measure some object's mass.
• First, we tried an iPhone, it comes out the period is 0.387s.
• By using the different values of Mtray, we get is weight between 162g and 164g.
• Then, we tried a tape. The period of it is 0.663s.
• Different values of Mtray show the mass is between 666g and 671g.

Conclusion: Today, we learned a new way to find the weight by measuring the period of shaking on the clamp. However, there are many sources may made uncertainty. Firstly, the detector may be not accurate enough. Secondly, calculation has some approximation. Thirdly, we do not really know the real weight of tray.